Characterisations and examples of graph classes with bounded expansion

Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Nesetril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several linear-time algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper, we establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters and the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with the Erdos-Renyi model of random graphs with constant average degree. In particular, we prove that for every fixed d>0, there exists a class with bounded expansion, such that a random graph of order n and edge probability d/n asymptotically almost surely belongs to the class. We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded non-repetitive chromatic number. We also prove that graphs with 'linear' crossing number are contained in a topologically-closed class, while graphs with bounded crossing number are contained in a minor-closed class.